Abstract
Discrete Liouville first passage percolation (LFPP) with parameter \(\xi > 0\) is the random metric on a sub-graph of \(\mathbb Z^2\) obtained by assigning each vertex z a weight of \(e^{\xi h(z)}\), where h is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all \(\xi > 0\). More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size \(2^n\) is typically at least \(2^{\alpha n}\) for an exponent \(\alpha > 0\) depending on \(\xi \). This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all \(\xi > 0\) and also has theoretical implications for the study of distances in Liouville quantum gravity.
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Notes
The reason for the factor of \(\pi /2\) here is to allow us to compare the discrete and continuum variants of LFPP with the same value of \(\xi \), c.f. [2]. This choice of constant makes it so that the variance of \(h_n(0)\) is asymptotic to \(\log (n)\).
LFPP with \(\xi =1/\sqrt{6}\) corresponds to Liouville quantum gravity with parameter \(\gamma =\sqrt{8/3}\) (equivalently, matter central charge \({\mathbf {c}_{\mathrm M}}= 0\)) and the fact that \(Q(1/\sqrt{6}) = 5/\sqrt{6}\) is a consequence of the fact that \(\sqrt{8/3}\)-LQG has Hausdorff dimension 4. See [10] for details.
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Acknowledgements
We thank an anonymous referee for helpful comments on an earlier version of the paper. We thank Josh Pfeffer for helpful discussions. J.D. was partially supported by NSF grant DMS-1757479. E.G. was supported by a Clay research fellowship and a Trinity college, Cambridge junior research fellowship. The research of A.S was supported by the ERC grant LiKo 676999 and is now supported by Grant ANID AFB170001 and FONDECYT iniciación de investigación \(\hbox {N}^o\) 11200085.
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Ding, J., Gwynne, E. & Sepúlveda, A. The distance exponent for Liouville first passage percolation is positive. Probab. Theory Relat. Fields 181, 1035–1051 (2021). https://doi.org/10.1007/s00440-021-01093-x
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DOI: https://doi.org/10.1007/s00440-021-01093-x